Finite-Dimensional CAR Algebra over an Inner-Product Space

Let $F$ be a finite-dimensional complex inner-product space, with the pairing linear in its first variable and conjugate-linear in its second. The algebraic CAR $\ast$-algebra over $F$, denoted ${\mathsf{C}\mathsf{A}\mathsf{R}}^{\mathrm{a}\mathrm{l}\mathrm{g}}(F)$, is generated by symbols $c(f)$, one for each $f\in F$, subject to complex linearity in $f$ and the relations

$\{c(f),c(g)\}=0,\{c(f{)}^{\ast },c(g{)}^{\ast }\}=0,\{c(f{)}^{\ast },c(g)\}=⟨g,f⟩1(f,g\in F),$

where $\{x,y\}=xy+yx$. Precisely, take the free unital complex $\ast$-algebra on symbols $c(f)$, with formal adjoints, and quotient by the two-sided $\ast$-ideal generated by these relations and the linearity relations. The quotient has the universal property that every complex-linear family $C(f)$ in a unital complex $\ast$-algebra satisfying the displayed relations induces a unique unital $\ast$-homomorphism from ${\mathsf{C}\mathsf{A}\mathsf{R}}^{\mathrm{a}\mathrm{l}\mathrm{g}}(F)$.

If $e_1,\dots ,e_n$ is an orthonormal basis of $F$, the elements $a_j:=c(e_j{)}^{\ast }$ and $a_j^{†}:=c(e_j)$ satisfy the finite-mode relations of Finite-Mode Complex CAR Algebra. The algebraic definition does not choose a basis; its finite-dimensional matrix realization is established in Finite-Dimensional CAR over an Inner-Product Space Is a Matrix Algebra.